#1
I have a continuous, one to one function f(x). How do I find g(x) such that g(g(x)) = f(x)?

Sorry, but such a function g (with the same domain and range as f) need not exist. Consider for example, the function , where . In this example it is easy to have an overview of all the possible Functions . Do you see one that satisfies for all ?

#2
Also, if i forever keep feeding the output of a continuous one to one funtion back into the input, isn't the result also continuous and one to one?

{i mean ...f(f(f(f(...f(x)))...}

How do find the values of x for which this converges to some finite value?

Can I find what the value is?

thanks

Well, if f is continuous and if the limit exists, it follows that such an x must be a fixed point of f: that is, x must satisfy the equation . In other words: to find those xs you solve the equation .